Method of playing board games on two-dimensional manifolds

ABSTRACT

A method of playing board games on modeled compact two-dimensional manifolds. The manifold is designed to be modeled from an imaginary game board by gluing its sides in the special manner so that the result is topologically equivalent to the desired surface as defined in topology.

COPYRIGHT STATEMENT

Copyright (c) Vladimir Patryshev, 2004

BACKGROUND OF INVENTION

In mathematics a manifold is a locally-Euclidean space; this inventionis limited to two-dimensional manifolds. These manifolds can be thoughtof as surfaces built from flat pieces by gluing them together alongtheir sides. See [2], [8] for definitions. Moebius tape, torus, Kleinbottle are examples of such surfaces (see [1]).

U.S. Pat. Nos. 6,595,519, 6,305,688, 6,318,726, 6,382,626, 6,491,300deal with puzzles that use Moebius topology. The current inventionrelates to playing board games on such a surfaces as well as otherpossible two-dimensional surfaces.

U.S. Pat. No. 4,026,557 relates to a group of games around Taurus game,that use the surface of torus to map its “oblique endless paths”.

One of possible games in Renju (see [4],[5] for the game rules), anotheris gomoku, or ‘five in line’ (see [6], [7] for the game rules). The game‘five in line’ (add the reference) is played on a two-dimensional boardconsisting of a rectangular grid of square cells. The game is played bytwo players (one uses ‘crosses’ and the other using ‘zeroes’ that inturn make their moves by drawing a ‘cross’ or a ‘zero’ in a cell thatwas not previously occupied by a ‘cross’ or a ‘zero’. The player thatmanages to have five adjacent ‘crosses’ or ‘zeroes’ in line, horizontal,vertical or diagonal, wins.

SUMMARY OF INVENTION

Board games can be played on virtual boards that are mapped totwo-dimensional surfaces like cylinder, torus, Klein bottle, etc, withthe necessary alteration of the rules. Such games can be played oncomputers or other digital devices, including game devices. In this casethe edges of the board are thought of as glued according to the gluingrules; the board can be scrolled on the screen, depending on how it isglued.

DETAILED DESCRIPTION

Usually board games are played on a limited (defined as compact inmathematics) piece of flat surface. Instead, the same games could beplayed on other surfaces e.g. a cylinder, a Moebius tape, a torus, aKlein bottle. While some of these surfaces can be physically constructedfrom a piece of flat material, others can be only modeled in a computeror on a flat board, if we define which sides should be imagined as gluedtogether.

For instance, to produce a cylinder (a) from a chess board, one has toglue A1 to H1, A2 to H2, etc., and A8 to H8. There is no necessity toactually do it, but the rules ought to be changed to take into accountthis imaginary gluing. This virtual gluing can be easily visualized on acomputer. To produce a Moebius tape (b), one has to glue A1 to H8, A2 toH7, etc., and A8 to H1. For torus (c), one has to take a cylinder (a),and glue together A1 and A8, B1 and B8, etc., and H1 and H8. For Kleinbottle (d) one has to take a cylinder (a), and glue together A1 and H8,B1 and G8, etc., and H1 and A8. This cannot be done in actual reality,but can be easily modeled on a computer or in the game rules.

Board games such as “renju” or “five in line” can be played on suchsurfaces. The recommended minimal size of the surface for “five in line”is 6 by 8, which is very practical for cell phones or PDAs. Both torusand Klein bottle surfaces can be used to play this variant of ‘five inline’.

A computer program, or a device (“apparatus”) can model any such gameboard surface and display it on the screen as if it were a flat surfacewith borders glued together according to the nature of the surface. Ifthe board does not have boundaries, e.g. a torus or a Klein bottle or aprojective plane, it can be panned or rotated on the screen, so thatdifferent cells move to the center of the screen. Moebius tape may berotated on the screen only in one dimension (left-right).

REFERENCES

-   1. http://en2.wikipedia.org/wiki/Klein_bottle-   2. http://en2.wikipedia.org/wiki/Manifold-   3. Rupert Matthews, “The Game Mania”, Silver Dolphin, 2002, ISBN    1571457070-   4. http://boardgames.about.com/cs/historiesnr/-   5. http://www.gamerz.net/pbmserv/renju.html-   6. http://www.gamerz.net/pbmserv/gomoku.html-   7. http://www.gamerz.net/pbmserv/pente.html-   8. R. Curant, H. Robbins, I. Stewart, “What Is Mathematics”, Oxford    University Press, 1996, ISBN: 0195105192571457070

1. A method of playing board games on models of arbitrary compacttwo-dimensional manifolds.
 2. The method according to claim 1 whereinthe manifold is a cylinder.
 3. The method according to claim 1 whereinthe manifold is a Moebius tape.
 4. The method according to claim 1wherein the manifold is a torus surface.
 5. The method according toclaim 1 wherein the manifold is a Klein bottle.
 6. The method accordingto claim 2, 3, 4, or 5 wherein the game is “renju” or “five-in-line”. 7.The method of rotating the board of methods 1 to 5 that places differentboard cells in the visible center of the board.
 8. The computer programproduct implementing methods declared in claims 1 to 7 wherein theplayer plays with a computer.
 9. The computer program productimplementing methods declared in claim 6 wherein the game is played ontwo or more portable computing, gaming or communication devicescommunicating via a network.
 10. A special device having a display and aprocessor, implementing the method of 6.